Difference between revisions of "Jacobi theta 3"

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$$\vartheta_3(z,q)=1+2\displaystyle\sum_{k=1}^{\infty} q^{k^2} \cos(2kz),$$
 
$$\vartheta_3(z,q)=1+2\displaystyle\sum_{k=1}^{\infty} q^{k^2} \cos(2kz),$$
 
where $\cos$ denotes the [[cosine]] function.
 
where $\cos$ denotes the [[cosine]] function.
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<div align="center">
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<gallery>
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File:Jacobitheta3,q=0.5plot.png|Plot of $\vartheta_3(z,\frac{1}{2})$.
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File:Complexjacobitheta3,q=0.5plot.png|Domain coloring of $\vartheta_3 \left(z,\frac{1}{2} \right)$.
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</gallery>
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</div>
  
 
=Properties=
 
=Properties=
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[[Logarithm of quotient of Jacobi theta 3 equals a sum of sines]]<br />
 
[[Logarithm of quotient of Jacobi theta 3 equals a sum of sines]]<br />
  
=See also=
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=References=
[[Jacobi theta 1]]<br />
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* {{BookReference|Special Functions|1960|Earl David Rainville|prev=Jacobi theta 2|next=Jacobi theta 4}}: $164. (3)$
[[Jacobi theta 2]]<br />
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* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Jacobi theta 2|next=Jacobi theta 4}}: $16.27.3$
[[Jacobi theta 4]]<br />
 
  
=References=
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{{:Jacobi theta footer}}
* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Jacobi theta 2|next=Jacobi theta 4}}: 16.27.3
 
  
 
[[Category:SpecialFunction]]
 
[[Category:SpecialFunction]]

Latest revision as of 20:23, 25 June 2017

Let $q \in \mathbb{C}$ with $|q|<1$. The Jacobi $\vartheta_3$ function is defined by $$\vartheta_3(z,q)=1+2\displaystyle\sum_{k=1}^{\infty} q^{k^2} \cos(2kz),$$ where $\cos$ denotes the cosine function.

Properties

Squares of theta relation for Jacobi theta 1 and Jacobi theta 4
Squares of theta relation for Jacobi theta 2 and Jacobi theta 4
Squares of theta relation for Jacobi theta 3 and Jacobi theta 4
Squares of theta relation for Jacobi theta 4 and Jacobi theta 4
Sum of fourth powers of Jacobi theta 2 and Jacobi theta 4 equals fourth power of Jacobi theta 3
Derivative of Jacobi theta 1 at 0
Logarithm of quotient of Jacobi theta 3 equals a sum of sines

References

Jacobi theta functions